Table 2.

Table of dimensionless variables and parameters.

IndependentDefinitionDimensionalizing
variablesScale
xMacroscale positionH
x~Mesoscale positionδ
tTime(Aτ0n))1
Dependent
variables
vVelocityAτ0nH
τStress tensorτ0
e˙Strain rate tensorAτ0n
ΩVorticity tensorAτ0n
ψStream functionAτ0nH2
ϕiVolume fraction phase in/a
kPhase field wave numberδ1
RiGrain size of phase i(B/(Aτ0n1))1/m
rInterface roughness(B/(Aτ0n1))1/m
VariousDefinitionNotes, equations 
factors
R0Reference grain size(76)
η0Twice basic state viscosity(79)
e˙0Basic state strain strain rate(80)
FFabric tensor(72)
ΓGary tensor(46)
ν, ΔνIsotropic, anisotropic viscosity contrasts(46)
ZZener pinning factor(73)
nInterface area factor3ϕ(1ϕ)
ParametersDefinitionValues used
CiGrain coarsening number, phase i0.2
CIInterface coarsening numberCi/2000
DiGrain-boundary damage number, phase i0.2-10
DIInterface damage numberDi/4
βDrip buouancy number3
χPhase mixing diffusivity108-104
pGrain coarsening exponent2
qInterface coarsening exponent4
nDislocation creep stress exponent3
mDiffusion creep grain size exponent3
cZener pinning coefficient0.87
IndependentDefinitionDimensionalizing
variablesScale
xMacroscale positionH
x~Mesoscale positionδ
tTime(Aτ0n))1
Dependent
variables
vVelocityAτ0nH
τStress tensorτ0
e˙Strain rate tensorAτ0n
ΩVorticity tensorAτ0n
ψStream functionAτ0nH2
ϕiVolume fraction phase in/a
kPhase field wave numberδ1
RiGrain size of phase i(B/(Aτ0n1))1/m
rInterface roughness(B/(Aτ0n1))1/m
VariousDefinitionNotes, equations 
factors
R0Reference grain size(76)
η0Twice basic state viscosity(79)
e˙0Basic state strain strain rate(80)
FFabric tensor(72)
ΓGary tensor(46)
ν, ΔνIsotropic, anisotropic viscosity contrasts(46)
ZZener pinning factor(73)
nInterface area factor3ϕ(1ϕ)
ParametersDefinitionValues used
CiGrain coarsening number, phase i0.2
CIInterface coarsening numberCi/2000
DiGrain-boundary damage number, phase i0.2-10
DIInterface damage numberDi/4
βDrip buouancy number3
χPhase mixing diffusivity108-104
pGrain coarsening exponent2
qInterface coarsening exponent4
nDislocation creep stress exponent3
mDiffusion creep grain size exponent3
cZener pinning coefficient0.87

Notes: The damage and coarsening numbers are extracted from the comparison to lab experiments by Bercovici et al. (2023), which are for lower pressure, higher temperature and much higher strain rates than in the mantle.

Table 2.

Table of dimensionless variables and parameters.

IndependentDefinitionDimensionalizing
variablesScale
xMacroscale positionH
x~Mesoscale positionδ
tTime(Aτ0n))1
Dependent
variables
vVelocityAτ0nH
τStress tensorτ0
e˙Strain rate tensorAτ0n
ΩVorticity tensorAτ0n
ψStream functionAτ0nH2
ϕiVolume fraction phase in/a
kPhase field wave numberδ1
RiGrain size of phase i(B/(Aτ0n1))1/m
rInterface roughness(B/(Aτ0n1))1/m
VariousDefinitionNotes, equations 
factors
R0Reference grain size(76)
η0Twice basic state viscosity(79)
e˙0Basic state strain strain rate(80)
FFabric tensor(72)
ΓGary tensor(46)
ν, ΔνIsotropic, anisotropic viscosity contrasts(46)
ZZener pinning factor(73)
nInterface area factor3ϕ(1ϕ)
ParametersDefinitionValues used
CiGrain coarsening number, phase i0.2
CIInterface coarsening numberCi/2000
DiGrain-boundary damage number, phase i0.2-10
DIInterface damage numberDi/4
βDrip buouancy number3
χPhase mixing diffusivity108-104
pGrain coarsening exponent2
qInterface coarsening exponent4
nDislocation creep stress exponent3
mDiffusion creep grain size exponent3
cZener pinning coefficient0.87
IndependentDefinitionDimensionalizing
variablesScale
xMacroscale positionH
x~Mesoscale positionδ
tTime(Aτ0n))1
Dependent
variables
vVelocityAτ0nH
τStress tensorτ0
e˙Strain rate tensorAτ0n
ΩVorticity tensorAτ0n
ψStream functionAτ0nH2
ϕiVolume fraction phase in/a
kPhase field wave numberδ1
RiGrain size of phase i(B/(Aτ0n1))1/m
rInterface roughness(B/(Aτ0n1))1/m
VariousDefinitionNotes, equations 
factors
R0Reference grain size(76)
η0Twice basic state viscosity(79)
e˙0Basic state strain strain rate(80)
FFabric tensor(72)
ΓGary tensor(46)
ν, ΔνIsotropic, anisotropic viscosity contrasts(46)
ZZener pinning factor(73)
nInterface area factor3ϕ(1ϕ)
ParametersDefinitionValues used
CiGrain coarsening number, phase i0.2
CIInterface coarsening numberCi/2000
DiGrain-boundary damage number, phase i0.2-10
DIInterface damage numberDi/4
βDrip buouancy number3
χPhase mixing diffusivity108-104
pGrain coarsening exponent2
qInterface coarsening exponent4
nDislocation creep stress exponent3
mDiffusion creep grain size exponent3
cZener pinning coefficient0.87

Notes: The damage and coarsening numbers are extracted from the comparison to lab experiments by Bercovici et al. (2023), which are for lower pressure, higher temperature and much higher strain rates than in the mantle.

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